I will be in Italy for a week on vacation next month. I believe the itinerary is Rome for 3 or 4 days, a day in Pompeii and a day or two on the Amalfi Coast.

I'm assuming that Rome is the place to look for work if that's even possible. I am an American with a recent CELTA certificate, a B.A. from a university Europeans will recognize and teaching and tutoring experience in the U.S. and Asia.

I am toying with the idea of approaching some schools with my credentials. How might I go about this?

As for the legalities of working in Italy, see the recent threads on this in the Italy forum.

As for the practicalities, now is not a good time to knock on any doors. Italians are gearing up for the long summer break and schools are winding down. Mid-September onwards is a much better time to look for work.

Enjoy your three days in Rome rather than rushing around trying to make appointments with people who probably don't care that much. See the sights, eat some ice-cream, and enjoy the Roman lifestyle as much as possible!

Ah, the Amalfi Coast. Fond memories - not of teaching English, mind. Schools will probably still be functioning admin-wise into July. But, as TiR stated, look at previous links for non EU folks trying to enter Italy. Try Salerno - easy access to the Amalfi Coast and Naples with great transport links for southern Italy. However, without a EU passport you may have problems. Enjoy la vacanza!